Question

Out of $8$ boys and $10$ girls, how many groups of $5$ boys and $6$ girls can be formed?

Answer 1

No. of ways grouping is possible.

There are $8$ boys and $10$ girls out of which $5$ boys and $6$ girls are to be selected at a time to form a group.

Therefore, the number of groups that can be formed of $5$ boys and $6$ girls are

$\begin{array}{c}{}^8{C}_5{\times }^{10}{C}_6=\frac{8!}{5!(8-5)!}\times \frac{10!}{6!(10-6)!}\\ =\frac{8!}{5!3!}\times \frac{10!}{6!4!}\\ =\frac{8\times 7\times 6\times 5!}{5!(3\times 2\times 1)}\times \frac{10\times 9\times 8\times 7\times 6!}{6!(4\times 3\times 2\times 1)}\\ =\frac{8\times 7\times 6}{3\times 2\times 1}\times \frac{10\times 9\times 8\times 7}{4\times 3\times 2\times 1}\\ =56\times 210\\ =11760\end{array}$

Hence, the number of groups can be formed of $5$ boys and $6$ girls out of $8$ boys and $10$ girls, are$11760$.